This invention relates to a magnetic shim for correcting a magnetic field so as to obtain a magnetic field of greater uniformity. More particularly but not exclusively, it relates to a magnetic shim for correcting the main magnetic field in an NMR (nuclear magnetic resonance) apparatus.
In an NMR apparatus, it is desirable that the main magnetic field be highly uniform in the region of an object which is being measured by NMR. One convenient and inexpensive method of correcting a magnetic field is the use of passive shims. A shim is a ring or cylinder of a magnetic material such as iron which is disposed in the main magnetic field of the NMR apparatus. The shim is magnetized by the main magnetic field and generates its own magnetic field which can compensate for a certain harmonic of the main magnetic field and increase the uniformity of the main magnetic field.
FIG. 1 is a longitudinal cross-sectional view of a portion of an NMR apparatus which employs a conventional iron shim. The NMR apparatus has a superconducting main coil 1 which generates a main magnetic field in the direction of the Z-axis. The main coil 1 is housed inside a cryostat 2 which includes a liquid helium tank or the like. The cryostat 2 has a room temperature bore 3 formed at its center into which an unillustrated object which is being measured is inserted. A pair of cylindrical iron shims 4 are coaxially disposed inside the bore 3 along the Z axis in symmetry with respect to an origin O. The Z coordinates of the end surfaces 4A and 4B of each shim 4 are Z.sub.1 and Z.sub.2, respectively. The angles .theta. between the Z axis and straight lines which connect ends 4A and 4B with the origin O are respectively .theta..sub.1 and .theta..sub.2, and the length of each iron shim 4 along the Z axis is L.sub.1.
Next, the operation of the iron shims 4 shown in FIG. 1 will be explained while referring to FIG. 2, which illustrates a portion of the iron shim 4 in an X-Z coordinate system. The case will be explained in which the iron shims 4 are cylindrical and are in a state of complete magnetic saturation due to the main magnetic field generated by the main coil 1. It will further be assumed that the component of the magnetic field due to magnetization (or the magnetic charge) of both end surfaces 4A and 4B is in the direction of the Z axis.
Normally, the main magnetic field which is generated by the main coil 1 contains, in addition to a zero-order harmonic B.sub.zo in the direction of the Z axis, a higher-order error field .DELTA.B.sub.z (X,Y,Z) which has a component .DELTA.B.sub.z (Z) in the direction of the Z axis. This Z-axis component is expressed by the following equation, in which C.sub.1, C.sub.2, C.sub.3, etc. are constants corresponding to higher-order harmonics: EQU .DELTA.B.sub.z (Z)=C.sub.1 Z+C.sub.2 Z.sup.2 +C.sub.3 Z.sup.3 +. . . (1)
As shown in FIG. 2, the magnetic field B.sub.z1 which is generated in the direction of the Z axis at the coordinate Z due to the magnetization J1 of one end surface 4A of the iron shim 4 is expressed by the following equation: EQU B.sub.z1 =B.sub.s .multidot.A.multidot.cos.theta..sub.1 /4.pi.r.sub.1.sup.2 . . . (2)
B.sub.s is the saturation flux density of the iron shim 4, A is the cross-sectional area of the end surface 4A, and r.sub.1 is the distance of the end surface 4A from the coordinate Z. The product (J.sub.1 A) of the magnetization J.sub.1 and the cross-sectional area A is referred to as the magnetic charge. In FIG. 2, only a portion of the iron shim 4 is illustrated. If the radius of the iron shim 4 is a, then the distance r.sub.1 from the coordinate Z and cos.theta..sub.1 are expressed by the following formulas: ##EQU1## Therefore, Equation (2) can be rewritten as follows: EQU B.sub.z1 =B.sub.s .multidot.A.multidot.(Z.sub.1 -Z){(Z.sub.1 -Z).sup.2 +a.sup.2 }.sup.-3/2 /4.pi. . . . (3)
In general, the main magnetic field at the coordinate Z can be expressed as a Maclaurin series. If Equation (3) is generalized and expressed as a Maclaurin series, then the magnetic field Bz at an arbitrary point Z which is generated by the magnetic charge on the end surface having a Z coordinate (Z.sub.1) is given by the following summation for k=0 to .infin.: EQU B.sub.z =.SIGMA.[(1/k!)(B.sub.z1 /.differential.Z.sup.k).sub.Z=0 ]Z.sup.k . . . (4)
Combining Equations (3) and (4) gives the following equation: ##EQU2## wherein B.sub.zs =B.sub.s .multidot.A/4.pi.a.sup.2. Furthermore, if EQU S=sin.theta.=a/R EQU U=cos.theta.=Z.sub.1 /R,
where R is the distance of the end surface 4A from the origin O, Then the coefficients .epsilon..sub.0, .epsilon..sub.1, etc. for each term are expressed as follows: EQU .epsilon..sub.0=S.sup.2 U EQU .epsilon..sub.1 =S.sup.3 (S.sup.2 -2U.sup.2) EQU .epsilon..sub.2 =(3/2)S.sup.4 U(-3S.sup.2 +2U.sup.2) EQU .epsilon..sub.3 =(1/2)S.sup.5 (-3S.sup.4 +24S.sup.2 U.sup.2 -8U.sup.4) EQU .epsilon..sub.4 =(5/8)S.sup.6 U(15S.sup.4 -40U.sup.2 S.sup.2 +8U.sup.4) etc. . . . (6)
As an example, the case will be considered in which the second-order harmonic of the main magnetic field, i.e., the harmonic which is proportional to Z.sup.2 is corrected. Each iron shim 4 is disposed so that the fourth-order harmonics of the magnetic fields which are generated by the surface magnetic charges which appear in end surfaces 4A and 4B are zero, so the following equation is satisfied: EQU .epsilon..sub.4 (.theta..sub.1)=.epsilon..sub.4 (.theta..sub.2)=0 . . . (7)
Combining Equations (6) and (7) gives EQU .epsilon.4(.theta.)=(5/8)S.sup.6 U(15S.sup.4 -40S.sup.2 U.sup.2 +8U.sup.4)=0
The two values of .theta. which satisfy this equation are EQU .theta..sub.1 .apprxeq.57.42.degree. EQU .theta..sub.2 .apprxeq.25.02.degree.
With these values of .theta., the relationship between the length L.sub.1 of the iron shim 4 in the direction of the main magnetic field (along the Z axis) and the radius a is L.sub.1 .apprxeq.1.5a.
Furthermore, if the magnetic field at the other end surface 4B is B.sub.z2, then from Equation (5), the second-order harmonic B.sub.z (Z.sup.2) of the magnetic field which is generated at the coordinate Z by both end surfaces 4A and 4B of the iron shim 4 is ##EQU3##
In Equation (8), the sign of .epsilon..sub.2 (.theta..sub.1) is negative because at end surfaces 4A and 4B, the poles are reversed with respect to one another.
When the iron shim 4 corrects the first-order harmonic B.sub.z (Z) of the main magnetic field, the end surfaces of the shims 4 are positioned such that the third-order harmonics of the magnetic fields which are generated at the coordinate Z by the surface magnetic charges in the end surfaces 4A and 4B are each 0, i.e., such that .epsilon..sub.3 (.theta..sub.1)=.epsilon..sub.3 (.theta..sub.2)=0. At this time, the length L.sub.1 of each shim 4 in the direction of the Z axis is L.sub.1 .apprxeq.1.32a. In addition, when the magnetic field in the X direction is corrected using iron shims which are divided in the circumferential direction, then L.sub.1 .apprxeq.1.15a.
It can be seen that conventional iron shims for correcting a magnetic field are quite long with respect to their radius. This results in the shims being very large and heavy, and due to their size, the support structure which resists the electromagnetic forces which are exerted on the shims must also be large.